. Hint:Use Problem 18 on Page 59 and if you like, you might use Pascal’s
triangle to construct the binomial coefficients.

If z,w are complex numbers prove zw =zw and then show by induction that
z1

⋅⋅⋅

zm =z1

⋅⋅⋅

zm. Also verify that ∑k=1mzk = ∑k=1mzk. In words this says the
conjugate of a product equals the product of the conjugates and the conjugate of a sum
equals the sum of the conjugates.

Suppose p

(x )

= anxn + an−1xn−1 +

⋅⋅⋅

+ a1x + a0 where all the ak are real numbers.
Suppose also that p

(z)

= 0 for some z ∈ ℂ. Show it follows that p

-
(z)

= 0
also.

I claim that 1 = −1. Here is why.

∘-----
2 √ --√ --- 2 √-
− 1 = i = − 1 − 1 = (− 1) = 1 = 1.

This is clearly a remarkable result but is there something wrong with it? If so, what is
wrong?

De Moivre’s theorem of Problem 4 is really a grand thing. I plan to use it now for
rational exponents, not just integers.

(1∕4) 1∕4
1 = 1 = (cos2π +isin2π) = cos(π ∕2)+ isin(π∕2) = i.

Therefore, squaring both sides it follows 1 = −1 as in the previous problem. What
does this tell you about De Moivre’s theorem? Is there a profound difference
between raising numbers to integer powers and raising numbers to non integer
powers?

Review Problem 4 at this point. Now here is another question: If n is an integer, is it
always true that

(cosθ − isinθ)

n = cos

(nθ)

− isin

(n θ)

? Explain.

Suppose you have any polynomial in cosθ and sinθ. By this I mean an expression of the
form ∑α=0m∑β=0naαβ cosαθ sinβθ where aαβ∈ ℂ. Can this always be
written in the form ∑γ=−

(n+m )

m+nbγ cosγθ + ∑τ=−

(n+m )

n+mcτ sinτθ?
Explain.

Does there exist a subset of ℂ, ℂ+ which satisfies 2.4.1 - 2.4.3? Hint: You might review
the theorem about order. Show −1 cannot be in ℂ+. Now ask questions about −i and i.
In mathematics, you can sometimes show certain things do not exist. It is very seldom
you can do this outside of mathematics. For example, does the Loch Ness monster exist?
Can you prove it does not?

Show that if a∕b is irrational, then

{ma + nb}

m,n∈ℤ is dense in ℝ. If a∕b is rational,
show that

{ma + nb}

m,n∈ℤ is not dense. Hint: From Theorem 2.15.1 there exist
integers, ml,nl such that

|mla + nlb|

< 2−l. Let Pl≡∪k∈ℤ

{k(mla+ nlb)}

. Thus this
is a collection of numbers which has successive numbers 2−l apart. Then consider
∪l∈ℕPl.